Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12963 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | #1867 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,22,28)(8,16,19,30,17,29,32,37)(9,13,21,27,20,26,33,36)(12,23,25,31,24,34,35,18);(6,8,10,19,11,17,22,32)(7,12,14,25,15,24,28,35)(9,18,21,23,20,31,33,34)(13,16,27,30,26,29,36,37);(8,17)(9,20)(12,24)(16,29)(19,32)(21,33)(25,35)(30,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5951);